Can I model with in a mode where quads are all planar?
I never needed that, but I think this workflow could do the job for you:

Creating new planar quads:


Make sure you have F2 addon enabled (it is enabled by default in Blender) by pressing CtrlAltU and navigating to Addons. Search for "F2" and enable it if it isn't already.
Extrude one vertex from existing polygon's edge so you get 2 edges prepared for the new quad
Select the middle vertex and move your mouse to the position you want to create a quad's opposite corner vertex
Press F - now the F2 addon creates a planar quad based on these 3 vertices






To move a vertex while keeping a polygon planar (when creating new polygons)


Make sure you have Transform Orientation set to Normal


Select a vertex and press G for move and then ShiftZ 2x to move on a plane perpendicular to Z normal of the face
This of course work only for corner vertices that are not part of other faces this is going to work when you are creating new polygons.
Alternatively, you can use Vertex Slide ShiftV that move's a vertex along edges so it also keeps a face planar. You can press Ctrl while sliding to be able to extend the sliding outside.






And as you have already noted, you can always press Spacebar > Make Planar Faces to convert the existing faces to planar ones.